Combinatorics of hexagonal fully packed loop configurations

TL;DR

This paper introduces hexagonal fully packed loop configurations (HFPLs), establishes necessary boundary conditions, and provides enumeration formulas using Littlewood-Richardson coefficients, advancing combinatorial understanding of these structures.

Contribution

It defines HFPLs, derives boundary condition inequalities, and enumerates them via Littlewood-Richardson coefficients, extending the combinatorial theory of loop configurations.

Findings

Necessary boundary conditions for HFPLs are established.

Enumeration of HFPLs is expressed through Littlewood-Richardson coefficients.

The number of HFPLs with specific boundary conditions is precisely counted.

Abstract

In this article, fully packed loop configurations of hexagonal shape (HFPLs) are defined. They generalize triangular fully packed loop configurations. To encode the boundary conditions of an HFPL, a sextuple $(\mathsf{l}_\mathsf{T},\mathsf{t},\mathsf{r}_\mathsf{T};\mathsf{r}_\mathsf{B},\mathsf{b},\mathsf{l}_\mathsf{B})$ of $01$-words is assigned to it. In the first main result of this article, necessary conditions for the boundary $(\mathsf{l}_\mathsf{T},\mathsf{t},\mathsf{r}_\mathsf{T};\mathsf{r}_\mathsf{B},\mathsf{b},\mathsf{l}_\mathsf{B})$ of an HFPL are stated. For instance, the inequality $d(\mathsf{r}_\mathsf{B})+d(\mathsf{b})+d(\mathsf{l}_\mathsf{B})\geq d(\mathsf{l}_\mathsf{T})+d(\mathsf{t})+d(\mathsf{r}_\mathsf{T})+\vert\mathsf{l}_\mathsf{T}\vert_1\vert\mathsf{t}\vert_0+\vert\mathsf{t}\vert_1 \vert\mathsf{r}_\mathsf{T}\vert_0+\vert\mathsf{r}_\mathsf{B}\vert_0\vert\mathsf{l}_\mathsf{B}\vert_1$ has to be fulfilled, where $\vert\cdot\vert_i$ denotes the number of occurrences of $i$ for $i=0,1$ and $d(\cdot)$ denotes the number of inversions. The other main contribution of this article is the enumeration of HFPLs with boundary $(\mathsf{l}_\mathsf{T},\mathsf{t},\mathsf{r}_\mathsf{T};\mathsf{r}_\mathsf{B},\mathsf{b},\mathsf{l}_\mathsf{B})$ such that $d(\mathsf{r}_\mathsf{B})+d(\mathsf{b})+d(\mathsf{l}_\mathsf{B})-d(\mathsf{l}_\mathsf{T})-d(\mathsf{t})-d(\mathsf{r}_\mathsf{T})-\vert\mathsf{l}_\mathsf{T}\vert_1\vert\mathsf{t}\vert_0- \vert\mathsf{t}\vert_1\vert\mathsf{r}_\mathsf{T}\vert_0-\vert\mathsf{r}_\mathsf{B}\vert_0\vert\mathsf{l}_\mathsf{B}\vert_1=0,1$. To be more precise, in the first case they are enumerated by Littlewood-Richardson coefficients and in the second case their number is expressed in terms of Littlewood-Richardson coefficients.